Abstract
Define the complexity of a regular language as the number of states of its minimal automaton. Let \( \mathcal{A} \) (respectively \( \mathcal{A}' \)) be an n-state (resp. n’-state) deterministic and connected unary automaton. Our main results can be summarized as follows:
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1.
The probability that \( \mathcal{A} \) is minimal tends toward 1/2 when n tends toward infinity
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2.
The average complexity of \( L{\text{(}}\mathcal{A}{\text{)}} \) is equivalent to n
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3.
The average complexity of \( L{\text{(}}\mathcal{A}{\text{)}} \cap L{\text{(}}\mathcal{A}'{\text{)}} \) is equivalent to \( \frac{{3\zeta (3)}} {{2\pi ^2 }}nn' \), where ζ is the Riemann “zeta”-function.
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4.
The average complexity of \( L{\text{(}}\mathcal{A}{\text{)}}^ * \) is bounded by a constant
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5.
If n ≤ n’ ≤ P(n), for some polynomial P, the average complexity of \( L{\text{(}}\mathcal{A}{\text{)}}L{\text{(}}\mathcal{A}'{\text{)}} \) is bounded by a constant (depending on P).
Remark that results 3, 4 and 5 differ perceptibly from the corresponding worst case complexities, which are nn’ for intersection, (n − 1)2 + 1 for star and nn’ for concatenation product.
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Nicaud, C. (1999). Average State Complexity of Operations on Unary Automata. In: Kutyłowski, M., Pacholski, L., Wierzbicki, T. (eds) Mathematical Foundations of Computer Science 1999. MFCS 1999. Lecture Notes in Computer Science, vol 1672. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48340-3_21
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